Analytical and finite element modelling of the dynamic interaction between off-road tyres and deformable terrains

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Analytical and Finite Element Modelling of

the Dynamic Interaction between Off-Road

Tyres and Deformable Terrains

by

Chrysostomos-Alexandros Bekakos

A thesis submitted in partial fulfilment of the requirements for the award of the Degree of Doctor of Philosophy

Department of Aeronautical and Automotive Engineering

Loughborough University

November 2016

Loughborough University, United Kingdom

Keywords: Tyre, Finite element method, tyre-soil interaction, off-road tyre, terramechanics

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This thesis is dedicated to my beloved parents Michael and Christina

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Abstract

Automotive tyres are one of the main components of a vehicle and have an extremely complex structure consisting of several types of steel reinforcing layers embedded in hyperelastic rubber materials. They serve to support, drive – accelerate and decelerate – and steer the vehicle, and to reduce transmitted road vibrations. However, driving is associated with certain types of pollution due to CO2 emissions, various particles due

to tyre wear, as well as noise. The main source of CO2 emissions is the tyre rolling

resistance, which accounts for roughly 30% of the fuel consumed by cars. The phenomenon becomes more pronounced in off-road conditions, where truck vehicles are responsible for about a quarter of the total CO2 emissions. Appropriate legislation

has been introduced, to control all of these pollution aspects. Therefore, tyre simulation (especially in off-road conditions) is essential in order to achieve a feasible design of a vehicle, in terms of economy and safety.

After a concise literature review and critical evaluation of the state-of-the-art models related to simulation and analysis of off-road tyres, the various limitations of the existing tyre models in terms of representing the rolling response and driving behaviour of actual tyres have been identified (e.g. utilization of non-invariant soil parameters). Finite element models for the terrain have been developed in which invariant soil parameters have been designated which are used for the description of the tyre – terrain interaction. Similar to the development of the soil models, a realistic tyre model was established via a novel coupled MATLAB – ABAQUS optimisation algorithm. The agreement of the tyre structure with reality was achieved through matching of its eigenproperties with analogous data from actual tyres. Subsequently, the interaction between a 235/75R17 tyre and a road – which is considered to be either rigid or deformable – was modelled with the finite element method and the rolling response of towed and driven wheels under various driving conditions was investigated. Regarding the limitations of the models used, it should be noted that the soil material is described by the linear Drucker-Prager constitutive model and the tyre parameters have been obtained via an optimisation procedure. More accurate soil constitutive models and calibration of their corresponding parameters, as well as

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realistic tyre properties can be used for further development of the various models involved in the thesis, the results of which can be validated with experimental data.

Additionally, a novel semi-analytical solution for the estimation of the response of a pneumatic tyre rolling on deformable terrain has been introduced, which involves substantial improvements compared to other existing semi-analytical solutions. Among others, lateral forces as well as the effects of treaded pattern and multi-pass have been taken into account. Although the developed analytical model is based on invariant soil parameters, it remains a semi analytical approach, as it involves empirical parameters such as the shear deformation modulus and empirical parameters related to distribution of the pressures between the tyre and the soil. Furthermore, it is assumed that the pressures at the tyre-soil interface are uniform along the width of the tyre which can lead to significant deviation of the results, especially for low inflated tyres (<15kPa) with large contact area.

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Acknowledgements

It is my pleasure to thank all of those who made this thesis possible. First and foremost I would like to thank my supervisors, Dr. Dan O’Boy and Dr. George Mavros for giving me the opportunity to undertake this research and also for their valuable help and guidance throughout the research period. Their friendship and advice both in technical and personal related issues are highly appreciated.

Secondly, I would like to thank Jaguar Land Rover and the UK-EPSRC grant EP/K014102/1 where as part of the jointly funded Programme for Simulation Innovation, I was able to investigate the very interesting topic of Terramechanics. Furthermore, I would like to thank my industrial partner, Jan Prins, who despite his heavy work load was always there for me and with his advice and help I was able to overcome many difficulties.

I would like to thank all of friends and colleagues at the Department of Aeronautical and Automotive Engineering, Loughborough University, for their support, help and resources required for the successful completion of this research. Special thanks go to Giancarlo Pavia, Agis Skarlas and Karol Bogdanski.

I would like to thank all of my friends and especially, Manolis Petrovitsos, Charis Akpinar, Kon/nos Thomopoulos, Christos Katrakazas and Leonidas Paouris for their help and support all of these years. Special thanks along with my sincere gratitude and highest appreciation goes to George Papazafeiropoulos, who apart from being an excellent colleague with remarkable research and critical skills was always there to support me and guide me throughout any difficulties I faced with personal and/or professional issues, thank you George.

Finally, and most of all, I would like to thank my family who has been always there to support me and encourage me to overcome any difficulty and made me believe that I can complete this academic route. I wouldn’t have done anything without them. From the bottom of my heart, thank you mom and dad.

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Nomenclature

a, a0, a1 Empirical constant [-]

a Half width defining the loading area, used in Eq. 2.16 [m]

a1 Angle of approach [o]

ar Cord orientation angle [o]

A0 Ageikin coefficient [-]

AR Cord area [m]

B, b Width of the plate or the wheel [m]

BH Ageikin parameter based on Bi [Pa]

Bi Soil bearing capacity [Pa]

Bn Mobility number as defined by Brixus(1987) [-]

CI Cone Index [-]

c Cohesion [Pa]

c1,c2 Empirical coefficients required for the determination of

the relative position of θΜ

[-]

Cb Number of penetrometer blows [-]

Cn Wheel numeric (dimensions must be selected such that the

wheel numeric is dimensionless)

[-]

C10,D1 Temperature dependent material parameters [-]

C1L,C2L,

C3L, C4L:

Pokrovsky’s theoretical values [-]

Cz Tyre Vertical Stiffness [N/m]

dDP Cohesion for Drucker-Prager [Pa]

D* Elastic tyre with bigger diameter [m]

D,d Diameter of the wheel [m]

Dh Hydraulic diameter of the contact area [m]

D1,D2,ξ,ω Parameters in Eq. (2.25) used as found on p.16-17 of

Lyasko (2010)

[-]

Dt Tyre damping coefficient [Ns/m]

e Tread height [m]

E Soil modulus of elasticity [Pa]

g Gravity constant [m/s2]

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fo Deflection of the elastic tyre [m]

fm, fk Dimensionless Kacigin’s friction coefficients of motion

and rest respectively

[-]

FMC Failure surface of MC model [Pa]

FDP Failure surface of DP model [Pa]

Fx Longitudinal Force [N]

Fz Vertical Force [N]

g Gravity constant [m/s2]

hL Height of the lug [m]

H0 Hardpan depth [m]

H1 New hardpan depth, Multi-pass model [m]

i,s Wheel slip/skid [-]

I1 First deviatoric strain [-]

J Ageikin coefficient [-]

j Shear displacement [m]

jx Longitudinal shear displacement [m]

jy Lateral shear displacement [m]

Jel Elastic volume ratio [-]

k Modulus of soil deformation, Bernstein-Goriatchkin [m/N1/2] kc Soil deformation modulus due to cohesive behavior [N/mn+1]

kφ Soil deformation modulus due to frictional behavior [N/mn+2]

kc ’

,kφ’ Dimensionless modulus of sinkage [-]

K1, K2 Parameters characterizing the shear stress-shear

displacement relationship

[-]

Kr Tyre sinkage ratio [-]

Ks Stiffness modulus of the terrain [Pa/m]

kG Equivalent static stiffness [N/m3]

kp Stiffness parameter for the soft substrate [N/m3]

kz Kacigin and Guskov’s coefficient of soil deformation [-]

kx Longitudinal shear deformation modulus [m]

ky Lateral shear deformation modulus [m]

Kc, Kγ,t Parameters in Eq. (2.10) used as on p.78 of Bekker(1960) [-]

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Kr Ratio of residual shear stress τr to the maximum shear

stress τmax

[-]

Kw Shear displacement where the τmax occurs [m]

L, l Length of the rectangular plate or the contact patch of the wheel

[m]

m Diameter exponent [-]

mm Strength parameter for the surface mat [N/m3]

MP Soil parameter that depends on moisture content [-]

n Exponent of deformation [-]

n0, n1 Sinkage exponent coefficients as used by Ding et al.(2014) [-]

Nγ,Nc, Nq Terzaghi’s parameters [-]

Nφ Flow value [-]

pgr Average ground pressure [Pa]

P Pressure [Pa]

q Bearing capacity of clay [Pa]

qf: Ultimate bearing stress [Pa]

qmax Ultimate bearing capacity [Pa]

qo Surcharge of the soil [Pa]

Q1 Ageikin parameter [-]

Qv Vertical load on wheel centre [N]

R,r Wheel radius [m]

Rb Resistance due to soil mass gathered in front of the wheel [N]

Rc Resistance due to compaction of the soil [N]

Rt Resistance due to tyre deformation [N]

Rtot Total Rolling resistance [N]

S Cord spacing [m]

t Width of tread contact area [m]

U Strain energy per unit of reference volume [-] Vt Tyre deformation velocity in the vertical direction [m/s2]

Vx Longitudinal velocity [m/sec]

W Vertical load [N]

z, s Tyre sinkage [m]

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zL Sinkage at the tip of the lug [m]

zr Rebound sinkage [m]

zT Depth of the track [m]

α,ε Parameters used as on Wong(2001), p.191 [-]

β Slip angle [o]

γ Soil unit weight [kg/m3]

γ1 New soil unit weight for multi-pass [kg/m3]

δt Tyre deflection [m]

θ Arbitrary angle along the wheel soil contact arc [rad]

θs, θ1, θ0 Entry angle of the wheel [rad]

θ2, θr: Exit angle of the wheel [rad]

θ4: Angle of transition for towed wheels [rad]

θΜ Angle where the maximum radial stress occurs [rad]

θe Entry angle for Gee-Clough model(90-θ1) [rad]

ρ Soil density [kg/m3]

σ1,σ2,σ3 Principal stresses [Pa]

τ, τj Shear stress [Pa]

φ Friction angle [o]

ψ Soil dilation angle [o]

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List of Figures

Figure 1.1.Flowchart of the research ... - 7 - Figure 2.1.Cone Penetrometer (Wong, 2001). ... - 10 - Figure 2.2.Dynamic behaviour of a rolling rigid wheel. ... - 14 - Figure 2.3. Soils’ Shear Stress response for three different types of soil, Wong(2001).

... - 19 -

Figure 2.4.Pneumatic Tyre on soft soil (Harnisch et al., 2005). ... - 28 - Figure 3.1. Average Von Mises stress for a Treadless Rigid plate interacting with a

deformable soil under 4kN of vertical load. ... - 43 -

Figure 3.2. Pressure Sinkage response of the soil for various normal pressures acting

on the plate ... - 45 -

Figure 3.3. Shear Stress developed on the plate – soil interface for various normal

pressures acting on the plate. ... - 45 -

Figure 3.4. Reference configuration used for the indentation process of a rigid wheel,

(a) Front view and (b) Right ... - 48 -

Figure 3.5. Reference configuration of the rolling rigid wheel model ... - 48 - Figure 3.6. Indentation model on cohesive soils: (a) Undeformed shape and (b)

deformed shape ... - 51 -

Figure 3.7. Indentation model on frictional soils: (a) Undeformed shape and (b)

deformed shape ... - 51 -

Figure 3.8. Dimensionless vertical load versus dimensionless sinkage for wheel with

b/d=0.3 on cohesive soil (φ=0o_{, ψ=0}o

and c/γgd=1.25). ... - 52 -

b/d=0.3 on frictional soil (φ=45o, ψ=0o and c/γgd=1.25 x 10-2). ... - 53 -

Figure 3.10.Dimensionless sinkage versus time for wheel with b/d=0.3 (φ=0o, ψ=0o, c/γgd=1.25) and various values of dimensionless vertical load (Qv/γbd2)... - 55 - Figure 3.11.Dimensionless steady-state sinkage for a wheel with aspect ratio b/d=0.3

(φ=0o_{, ψ=0}o

and c/γgd=1.25). ... - 55 -

Figure 3.12.Dimensionless sinkage versus time for various aspect ratios of the wheel

rolling on soil with φ=0o_{, ψ=0}o

and c/γgd=1.25 and Qv/γbd2=1.9. ... - 57 - Figure 3.13.Dimensionless sinkage versus time for various aspect ratios of the wheel

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Figure 3.14.Wheel with b/d=0.3 (φ=0o, ψ=0o, c/γgd=1.25, Qv/γbd2=2.4): (a) Direction

of travel from left to the right and (b) front view of the wheel. ... - 58 -

Figure 3.15. Dimensionless sinkage versus time for a rolling wheel with b/d=0.3,

φ=45o_{, c/γgd=0.25, Q}

v/γbd2=1.9 for various values of the soil dilation angle (degrees).

... - 59 -

Figure 3.16. Treaded wheel with b/d=0.3rolling on cohesive soil (φ=0o, ψ=0o, c/γgd=1.25): (a) Purely longitudinal tread pattern and (b) purely lateral tread pattern. - 60 -

Figure 3.17. Ratio of required horizontal force to vertical load of rolling wheel with

b/d=0.5 (φ=45o_{, c/γgd=0.25) for various combinations of longitudinal tread ratio e/t}

and dimensionless vertical load Qv/γbd2. ... - 60 - Figure 3.18. Slip ratio of rolling laterally treaded wheel with b/d=0.5 (φ=45o, ψ=0o, c/γgd=0.25) versus the lateral tread ratio e/d for various lateral tread ratios t/d. ... - 61 -

Figure 4.1. Tyre Structure (http://www.avtogumi.com/en/polezno/struktura.php).- 65

-

Figure 4.2.Tyre half-cross section geometry. ... - 68 - Figure 4.3.Illustration of the tyre model. ... - 72 - Figure 4.4.Inner components of the detailed 3D FE tyre model. ... - 73 - Figure 4.5. Eigenmode shapes of the optimised tyre model (continued in the next

page). ... - 77 -

Figure 4.5. Eigenmode shapes of the optimised tyre model (continued from previous

page). ... - 78 -

Figure 4.6. Eigenfrequencies of various mode shapes of the optimised tyre model for

various inflation pressures. ... - 79 -

Figure 5.1.Procedure of development and analysis of the various models used in this

study. ... - 85 -

Figure 5.2. Mesh configuration of the tyre used (a) for the steady state transport

analysis and (b) for the transient dynamic analysis in this study. ... - 87 -

Figure 5.3. Results of footprint analysis for model in Figure 5a: vertical deflection

versus vertical load for various inflation pressures. ... - 87 -

Figure 5.4. Results of footprint analysis for model in Figure 5a: contact area versus

vertical load for various inflation pressures. ... - 88 -

Figure 5.5.Deflection versus vertical load for various orientations of the rebar of the

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Figure 5.6.Vertical deformation of tyre in the static footprint analysis, with 242kPa

inflation pressure and 5kN vertical load. ... - 89 -

Figure 5.7.Vertical deformation of tyre in the static footprint analysis, with 160kPa

inflation pressure and 5kN vertical load. ... - 89 -

Figure 5.8. Results of SST analysis for model in Figure 5.2(a): rolling resistance

force versus angular velocity. ... - 91 -

Figure 5.9. Results of SST analysis for model in Figure 5.2(a): torque versus angular

velocity for various vertical loads. ... - 91 -

Figure 5.10. Results of SST analysis for model in Figure 5.2(a): identification of free

rolling conditions. ... - 92 -

Figure 5.11.Transient rolling process of a wheel impacting on a rigid bump. ... - 92 - Figure 5.12.Effect of inflation pressure on the vertical displacement of the centre of

the tyre during and after its impact with the bump. ... - 94 -

Figure 5.13.Effect of vertical load on the vertical displacement of the centre of the

tyre during and after its impact with the bump. ... - 95 -

Figure 5.14. Vertical response of the spindle due to rigid bump impact: effect of free

rolling velocities on displacement. ... - 97 -

Figure 5.15. Vertical response of the spindle due to rigid bump impact: effect of free

rolling velocities on acceleration, for a tyre with vertical load 5kN and inflation pressure 200 kPa. ... - 97 -

Figure 5.16.Reference configuration of the model (b) of this study. ... - 98 - Figure 5.17. Angular velocity of the towed wheel considered in this study rolling on

soft soil (c=1.25γgd, φ=0, ψ=0): effect of linear velocity. ... - 102 -

Figure 5.18. Angular velocity of the towed wheel considered in this study rolling on

soft soil (c=1.25γgd, φ=0, ψ=0):effect of vertical load... - 102 -

Figure 5.19. Angular velocity of a towed wheel rolling on soft soil (c=1.25γgd, φ=0,

ψ=0) versus horizontal travelling distance for various inflation pressures. ... - 103 -

Figure 5.20.Deformed geometry of the tyre-soil system. ... - 104 - Figure 5.21.Cross section of the simplified deformable tyre (half-axisymmetric

model). ... - 105 -

Figure 5.23.Deformed configuration of the simplified FE tyre – soil model. ... - 106 - Figure 5.24. Slip ratio of the tyre versus its inflation pressure, for b/d=0.33 and

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Figure 5.25. Dimensionless sinkage on soft soil (c=1.25γgd, φ=0, ψ=0) with ω=7.46

rad/s and Qv=1.2γgbd2 for a wide range of inflation pressures. ... - 109 -

Figure 5.26. Linear velocity of a wheel on soft soil (c=1.25γgd, φ=0, ψ=0) with

ω=7.46 rad/s and Qv=1.2γgbd2 for two distinct inflation pressures. ... - 110 -

Figure 5.27. Dimensionless sinkage of a wheel on soft soil (c=1.25γgd, φ=0, ψ=0)

with ω=7.46 rad/s and inflation pressure 242 kPa for two values of the vertical load. . - 111 -

Figure 6.1. Coupled Eulerian Lagrangian (CEL) model, average Von Mises stress for

(a) un-deformed configuration, (b) deformed configuration with void elements and (c) without void elements, for a cohesive soil (φ=0o_{, ψ=0}o

and c/γgd=1.25). ... - 116 -

Figure 6.2.Dimensionless vertical load versus dimensionless sinkage for wheel with

b/d=0.3 and cohesive soil (φ=0o_{, ψ=0}o

and c/γgd=1.25). ... - 117 -

Figure 6.3. Average Von Mises stress of the CEL model of a rigid rectangular plate

with dimensions 0.15m by 0.3048m indented into a cohesive soil (φ=0o_{, ψ=0}o

and c/γgd=1.25), (a) Side view and (b) Top view of the reference configuration. ... - 118 -

Figure 6.4.Fitting of LSA model to numerical pressure-sinkage response for frictional

sand. ... - 119 -

Figure 6.5.Pressure-Sinkage response for various soils, using LSA model. ... - 121 - Figure 6.6.Shear Stress response for frictional sand, as per Janosi-Hanamoto’s model.

... - 122 -

Figure 6.7.Pressure vs sinkage for a rectangular plate of various sizes interacting with

frictional sand, using LSA model. ... - 123 -

Figure 6.8.Static indentation of a rigid wheel. ... - 125 - Figure 6.9.Effect of slip ratio on the dynamic sinkage. ... - 126 - Figure 6.10.Reference configuration for a driven wheel rolling on a soft soil. ... - 127 - Figure 6.11.Flowchart of the semi-analytical procedure for estimation of tyre-soil

interaction forces. ... - 128 -

b/d=0.3 on cohesive soil. ... - 129 -

b/d=0.3 on frictional soil. ... - 130 -

Figure 6.14. Dimensionless vertical load versus dimensionless steady state sinkage

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Figure 6.15. Drawbar pull developed for a rigid wheel rolling on wet clay with

various vertical loads, versus its slip ratio. ... - 132 -

Figure 6.16. Drawbar pull developed for two rigid wheels of different width, rolling

on wet clay with vertical load equal to Fz=4kN, versus slip ratio. ... - 133 - Figure 6.17. Drawbar pull developed for three commercially available tyres rolling

on wet clay with vertical load equal to Fz=4kN, versus slip ratio. ... - 134 - Figure 6.18. Drawbar pull developed for Wheel 3 rolling on three deformable soils

with vertical load equal to Fz=2kN, versus slip ratio. ... - 135 - Figure 6.19. Total sinkage versus slip ratio for a rigid wheel with dimensions

b=0.215m, D=0.8728m and vertical load Fz=4kN, for two different types of soil.- 136

-

Figure 6.20. Lateral Force developed for Wheel 3 rolling on three deformable soils

with slip ratio equal to 0.2 and vertical load equal to Fz=2kN versus slip angle. . - 138 - Figure 6.21. Lateral Force versus slip angle developed for Wheel 3 rolling on clay,

for slip equal to 0.2 and various vertical loads. ... - 139 -

Figure 6.22.Schematic representation of a tread block between a treaded tyre and the

soil. ... - 140 -

Figure 6.23. Drawbar pull versus slip ratio developed for a treadless and a treaded

rigid wheel, rolling on moist loam with vertical load equal to Fz=10kN... - 141 - Figure 6.24. Drawbar pull versus slip ratio developed for a rigid wheel with different

void ratio, rolling on moist loam with vertical load equal to Fz=10kN. ... - 142 - Figure 6.25.Reference configuration of a pneumatic tyre, and the equivalent

substitute circle. ... - 143 -

Figure 6.26. Drawbar pull developed for a pneumatic tyre with different inflation

pressures, rolling on moist loam with vertical load equal to Fz=4kN, versus slip ratio. -

144 -

Figure 6.27. Drawbar pull developed two pneumatic tyres, rolling on moist loam with

vertical load equal to Fz=4kN, versus slip ratio. ... - 145 - Figure 6.28.Schematic representation of the multi-pass effect. The rear wheel rolls

over the exact same rut path with the front wheel. ... - 147 -

Figure 6.29.Drawbar pull developed for a front and rear wheel under the assumption

of rolling on the exact same rut path for a vertical load of 4kN and 0.2 slip, rolling on wet clay. ... - 148 -

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Figure A.1.Dimensionless sinkage versus time for various aspect ratios of the wheel

and c/γgd=1.25 and Qv/γbd2=1.9. ... - 169 - Figure A.2.Dimensionless sinkage versus time for various aspect ratios of the wheel

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Chapter 1 Introduction

1.1. Introduction

There are many instances in which an off-road vehicle is required to travel on deformable terrain, such as space missions, agriculture, construction, etc, in which cases realistic off-road tyre models are needed for the prediction of the interaction of the vehicle with the supporting terrain. However, little research has been done in this area, since the majority of research in tyre development has concentrated on on-road vehicles. In this chapter the context of the research is presented followed by the aim and the desired objectives. Next, the novelties and contributions of this study are briefly presented along with a short summary of the subsequent chapters.

Since the computational power of modern computers is continuously growing, simulations are becoming essential tools for mechanical engineers. In this aspect, vehicle manufacturers are funding projects where full vehicle models, capable of assessing a vehicle design prior to its production, are developed. Tyre as one of the most important components of an automobile constitutes the main link between the vehicle and the ground and is mainly responsible for the driving response of the vehicle under accelerating/braking and steering conditions; therefore, accurate and realistic tyre models – which can then be integrated into full vehicle models – are necessary.

1.2. Project Context

This work was supported by Jaguar Land Rover and the UK-EPSRC grant EP/K014102/1 as part of the jointly funded Programme for Simulation Innovation. The help and guidance provided by Jaguar Land Rover are highly appreciated and were crucial for the completion of this project.

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In many cases there is the necessity for construction of vehicles which are capable of traveling on off-road terrains. Therefore, efficient simulation of the dynamic interaction between off-road tyres and deformable terrains is of high importance. Among the various applications of off-road vehicles, the latter are used in agriculture operations, where economy during excavations for tillage is desired, and potential immobilization of the vehicles has to be avoided. In addition, military tracked and wheeled vehicles are necessary in the national defence of each country, and to maximize their potential, they have to be designed for optimum performance during travelling on soft terrain and/or snow. Furthermore, off-road vehicles are needed in various construction sites, where pavements and/or rigid foundations are not present. In this case, the vehicles need to be appropriately immobilized to ensure safety and to ensure that the construction works are carried out properly. Commercial off-road vehicles are also needed for cross-country transportation and racing (e.g. rally or desert racing), where it is obvious that they have to perform optimally on non-homogeneous terrain, such as rocky and granular soils.

On-road tyres have attracted significant more attention compared to off-road tyres in the past years and for that reason the majority of the existing off-road tyre models are usually utilizing simplistic empirical and/or semi-analytical equations with inherent limitations and a number of restrictive assumptions. For instance, use of non-invariant soil parameters into tyre models (e.g. Bekker, 1956), necessitates continuous soil experimental measurements which in return increase the overall cost. Furthermore, such models can be applied only for a limited range of tyre geometries, where the width of the tyre has to be sufficiently large in order to avoid significant deviations from experimental measurements (Meirion-Griffith & Spenko, 2010); therefore the utilization of such models in lunar rovers, where narrow wheels are predominantly used, would yield inadequate results. Another major assumption commonly used in the off-road tyre mechanics is that of a constant pressure distribution along the width of the wheel, which often tends to underestimate or overestimate the traction response of the tyre. Finally, in the majority of empirical tyre models, the dynamic sinkage caused by the slip/skid rolling conditions of the tyre is either not reflected, or taken into account in a very simplistic way (Lyasko, 2010b).

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It is expected that the results of this thesis will be used for the establishment of guidelines and relevant methodologies for off-road tyres rolling on soft terrains intended to minimize their production cost and maximize their rolling performance.

1.3. Aim

The purposes of this thesis are the following:

 Understanding the current status of off-road tyre model development and identification of the limitations present in such models. Based on these, it is easier to decide where additional knowledge is necessary in terms of improving the existing models.

 Determination of the most important tyre design parameters with regards to off-road tyre locomotion, such as tyre width, inflation pressure, tread pattern/void ratio and detailed geometric characteristics of the tyre structure. The latter, apart from providing a conceptual description of this phenomenon, will also help other researchers in their future studies to focus on specific parameters which have the largest impact on the tyre response and reduce unnecessary complexity.

 Improvement of existing models (analytical and numerical) so that they provide a better description of the phenomena of static and dynamic off-road tyre – terrain interaction. Attempts will be made towards the exclusion of non-invariant soil parameters from the analytical off-road tyre modelling techniques and the inclusion of tyre structure details in the numerical models, such as cord orientation and reinforcing layers’ thicknesses.

 To develop efficient optimization techniques for the parameterization of both the soil and the tyre.

1.4. Objectives

This research is mainly focused on the development of novel tyre-terrain models, both numerically and analytically, in order to accurately calculate the dynamic response of an off-road tyre by attempting to eliminate a number of limitations associated with the current status of off-road tyre modelling. Reliable numerical modelling of off-road tyres is vital for obtaining realistic results which are used for the design.

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 Development of numerical tyre-terrain models using the commercial finite element code Abaqus v.6.13. Both rigid and pneumatic tyres, either driven or towed, interacting with various types of deformable soils will be modeled in order to improve existing models and provide a better understanding of the static and dynamic off-road tyre – terrain interaction.

 Development of a semi-analytical tyre-terrain model using the programming language MATLAB in order to identify how the compliance of the tyre, its geometrical characteristics and dynamic loading effects (such as rolling velocity and vertical load) affect the rolling response of the tyre on terrains with various bearing capacities.

 Both numerical and analytical models will be used to carry out parametric studies in order to determine the effect of various design, operational and environmental (soil) parameters in tyre behavior.

 Finally, one of the main objectives with principal interest for this thesis is the identification of the most important invariant soil parameters which determine the tyre behaviour, which is governed to a large extent by the alteration of the soil properties due to vehicle loads. It is apparent that soil compaction caused by a rolling wheel affects its structure, decreases its porosity and water and air infiltration, (e.g. reduces crop yield which is caused by hindering of root penetration). Following that, the energy efficiency of an off-road tyre in terms of rolling resistance and inflation pressure will be addressed.

1.5. Research Contribution

A robust methodology regarding the development of a valid FE tyre – terrain model has been presented, which involves two different FE models: (a) the soil model and (b) the tyre model. With regards to the soil model, a rigid wheel – deformable terrain coupled model has been developed in order to assess the accuracy and robustness of the models involved. A novel equation for conversion of the Mohr-Coulomb to Drucker-Prager soil model and vice-versa has been developed for triaxial tension or compression.

Following that, a pneumatic tyre model has been developed and the natural frequencies of the tyre structure have been extracted. A novel coupling MATLAB – ABAQUS optimisation technique for tyre development has been proposed where the

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geometrical characteristics with high impact on the behaviour of the tyre, such as the thickness of the belt and/or the orientation of the cords, are the design variables, and the objective function describes how well the natural frequencies of the numerical tyre model and those obtained from experimental testing found in the literature are matched.

With regards to the analytical formulations, the majority of the existing semi-analytical off-road tyre models are based on empirical material laws (Bekker, 1956 & Reece, 1965) which use non-invariant soil parameters such as kc and kφ. In the current

study Lyasko’s (2010a) analytical mathematical model for describing the rigid plate – soil indentation is utilised and modified accordingly in order to incorporate the geometry of rolling wheel being either rigid or pneumatic. Thereafter, a novel semi-analytical tyre model has been developed with the use of four invariant soil parameters, namely as the cohesion, the friction angle, the soil unit weight and the Young modulus. These soil parameters can be easily measured in-situ with hand held instruments like a bevameter or a cone penetrometer. Furthermore, the slip sinkage effect has been incorporated in the model where with every increase on the slip conditions of the wheel, there is an increase in the vertical displacement of the wheel into the soil, capturing the digging effect.

Due to the existing lack in the literature of studies containing models that use realistic invariant soil parameters and tyre physical properties, in this thesis the models developed were verified in terms of their qualitative response; therefore further validation studies are necessary in order to establish the level of quantitative agreement with measurements.

1.6. Outline

Chapter 2 presents a thorough literature review and critical assessment on the state-of-the-art techniques with regards to the existing empirical, analytical and numerical methods for the assessment of the off road vehicle performance. In addition to that, techniques of off-road tyre modelling which incorporate the aforementioned methods have been reported. Literature finding are critically evaluated and the aims/objectives of the work are revisited.

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Chapter 3 introduces the FE modelling technique based on which the soil properties used for the various models in this study have been established. Dimensionless graphs have been produced and the validity of the soil model has been confirmed with comparison of experimental results found in the literature.

Chapter 4 focuses on the development of a realistic numerical tyre model via a novel coupled MATLAB – ABAQUS optimisation algorithm. The eigenfrequencies and eigenvalues of the tyre structure have been extracted and their variation with increasing inflation pressure has been illustrated.

Chapter 5 incorporates the two aforementioned FE models into a single FE model capable of predicting the off road performance of a realistic 235/75R17 tyre. The response of the pneumatic tyre in contact with a rigid surface, for a number of inflation pressures and vertical loads has initially been examined, and results of the contact area and the vertical deflection, measured from the centre of the wheel, have been presented. Following that, the response of the rim for a pneumatic tyre rolling over a speed bump has been illustrated. Furthermore, towed and driven wheels interacting with cohesive and frictional deformable terrains have been modelled and the effect of various parameters such as the inflation pressure, on the overall driving response has been presented.

Chapter 6 introduces a novel semi-analytical tyre model capable of quantitatively capturing the realistic response of a pneumatic tyre rolling on a deformable terrain. The proposed equation utilizes invariant soil parameters and is derived from soil mechanics theory. Initially, a rigid plate is forced into the soil and the pressure-sinkage response is presented according to Lyasko’s (2010a) equations for a number of different soils. Following that, a rigid wheel has been modelled and the effects of the vertical load, the width of the tyre and the tread pattern on the overall driving response have been illustrated. In addition, a pneumatic tyre has been modelled under the same concept with the one initially proposed by Bekker (1956) and further implemented by Harnisch et al. (2005). Finally, the multi-pass effect has been modelled in a similar manner with the one presented by Harnisch et al. (2005) under the assumption that the rear wheel rolls over the exact same rut path created by the front wheel.

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Finally, Chapter 7 summarizes the results drawn from the current study and focuses on recommended future work for further implementation and validation of the developed models. An outline of the research is presented in Fig. 1.1.

Figure 1.1.Flowchart of the research Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7

Review of Literature Shortcomings and Limitations Identified

FE soil modelling

FE tyre modelling

FE tyre-terrain model

Stiff Soil

Soft Soil

Boundary between Stiff and Soft Soil

Semi-Analytical Model θM θr θ1 zr ω R θ0 θ2 Vx W C zdynamic A B DP

Conclusions & Future Work

Inve sti g a ti on of the pa ra mete rs with pr incipa l e ffe c t on the rollin g re sponse

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Chapter 2 Review of Literature

2.1. Introduction

This chapter presents the available methods and techniques regarding the assessment of a vehicle performance on off road conditions. Initially, the empirical methods related to the characterization of the strength of the soil, the pressure sinkage and/or the shear stress-shear displacement responses are presented, followed by the equivalent analytical and numerical formulations.

Terramechanics is the field of science that deals with the interaction of a vehicle with the underlying deformable soil. Understanding the underpinning principles by which tyre forces, such as rolling resistance and drawbar pull, are developed will encourage the manufacturing of optimal off road tyres. In addition, this understanding will assist on the prediction of vehicle performance on soft soils. Whilst, the interaction between on road tyres and urban pavements has received much attention and has been subjected to significant research in the past, the same type of interaction in off road conditions has not been adequately represented to a similar extent in terms of analytical and/or numerical methods. This lack is mainly based on the multivariable and complex nature of the physical interaction of an off road tyre with the underlying soft soil. For instance, a simplified dynamic off road tyre-soil interaction will be represented by the summation of the movement of the particles of the soil and the deformation of the tyre as the total deformation; while for the on road tyre community and for a simplified on road tyre model, since the pavement is regarded as a rigid surface, the total deformation is concentrated only on the tyre interface.

Given the complexity of tyre-soil interaction several assumptions have to be made in terms of creating accurate and yet computationally efficient methods. These assumptions may range from a simplified linear soil material response, such as a purely cohesive and/or a purely frictional soil, up to a rigid wheel tyre response where

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a highly inflated tyre (inflation pressure larger than 250kPa) is assumed to roll over a relatively soft soil. To this regard, various methods have been developed, where the ability of tracked and/or wheeled vehicles to roll over a deformable terrain can be feasibly characterized.

2.2. Empirical Methods

2.2.1 Soil assessment

Identifying the principal characteristics regarding the response of the terrain, such as cohesion and friction angles, under normal loading and shear stress is the primary focus in the field of Terramechanics. The most pronounced methods with regards to the terrain classification, involve hand held instruments and techniques like: i) the cone penetrometer, ii) the bevameter and iii) traditional soil mechanic techniques used in the field of civil engineering (Wong, 2001). Following that, system metrics such as, the Vehicle Cone Index (VCI), the Mobility Index (MI) and the Mean Maximum Pressure (MMP) have been developed (Priddy & Willoughby, 2006), which permit a long term characterization of the ability of the vehicle to roll over specific types of terrains.

VCI is defined as the minimum soil strength, necessary for a self-propelled vehicle, to traverse a certain type of soil for a prescribed number of times without getting immobilized (VCI1 and VCI50). Numerous empirical equations have been developed

through which VCI can be measured with the use of easily captured parameters, like the weight of the vehicle. With regards to correlating vehicle characteristics with the VCI’s value, the principal parameter MI was developed. Based on that value the locomotion of the vehicle can be assessed. At the same time, the United Kingdom’s Ministry of Defence used a different index (MMP) to characterize the traversability of a vehicle. Herein, a common misconception should be clarified since the MMP should not be compared with the VCI, but instead with the MI. This is due to the fact that the former is a performance metric and not a set of predictive equations (Priddy & Willoughby, 2006).

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Furthermore, the cone penetrometer was developed by the Waterways Experiment Station (WES) of the US Army Corps of Engineers as a hand held device, consisting of a 30 degree circular cone with a base area of 3.23 cm2, Fig. 2.1. By utilizing this device, the parameter so called, cone index, which represents a combination of the shear and compressive characteristics of the soil may be obtained. This technique was developed as “go/no go” device during the Second World War, with regards to the assessment of a vehicle’s capacity to roll over a certain terrain without being immobilized. However, ambiguous opinions exist to whether this device can adequately assess the aforementioned potential for immobilization such as work conducted by Reece & Peca (2006). They state that, the latter device can successfully capture the response of frictionless clay but remains still inadequate to characterize the properties of sand. Therefore, the necessity for of a handheld device and/or a set of equations capable of predicting the behavior of both cohesive and frictional terrains is apparent.

Figure 2.1.Cone Penetrometer (Wong, 2001).

Based on the resulting measurement of the cone penetrometer (Cone Index, CI), several techniques and/or methods have been developed for vehicle performance assessment. Wismer & Ruth (1973) proposed Eq. 2.1 as a function of CI where the first term is the gross traction and the second term is the resistance. However, the slip sinkage effect was not included in the above-stated model and, for this reason, Brixius (1987) further developed Eq. 2.1 into Eq. 2.2. Brixius’ empirical model is extensively used on off-road vehicle simulations. Following Eq. 2.1 and 2.2, similar empirical models have been developed (see Grisso et al., 2006) in an attempt to correlate vehicle performance with CI. However, currently, empirical models which only depend on CI values are considered to be deficient in vehicle performance

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measurements since it is well established that soil characteristics, like compactness and hardness, cannot be correlated with CI measurements.



0.3 C Sn



n P 1.2 0.75 1 e 0.04 W C     _ _      (2.1)



0.1 Bn





7.5 i



Q 0.88 1 e 1 e r W 0.04   _   _      (2.2)

The bevameter technique developed by Bekker (1957) is based on the hypothesis that terrain characteristics relevant to off road locomotion are best measured under the same loading conditions as those exerted by an off-road vehicle. In this respect, two separate tests have been developed. The first test refers to the evaluation of the pressure-vertical displacement equation, when a certain pressure, similar to that exerted by vehicles, is acting on a simulation plate with a size similar to that of the contact patch of a vehicle’s running gear. In the second test, the shear stress - shear displacement equation, presented in the following sections, is considered under multiple normal loads, thus measuring parameters like the tractive effort or the slip characteristics of a vehicle. Henceforth, the vertical displacement caused by the running gear (rigid wheel, pneumatic tyre) into the soil will be referred to as sinkage which is the main term used in the field of Terramechanics.

Additionally, classic soil characterization methods have been developed by civil engineers, whereby soil samples are taken from the field and tested in the laboratory with the use of devices such as a triaxial apparatus and/or a direct shear box. Gan et al. (1988) preferred the shear box technique over the triaxial apparatus (since less time is required to fail the specimen) and several tests have been performed on unsaturated soil. However, it can be argued that the use of the cone penetrometer and the bevameter constitutes a more realistic approach since the soil is at its natural state, while a civil engineering method would necessitate the disturbance of the terrain for the sampling process.

2.2.2 Pressure – Sinkage Equation

To the best of the author’s knowledge, regardless of the assumptions involved, in most of the models published in the literature the off-road tyre-soil interaction is studied in terms of two main effects. The first is responsible for the relationship

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between the vertical load exerted by the vehicle and the respective sinkage of the wheel into the soil; the second, deals with the shear stress-shear displacement developed at the tyre-soil interface due to tyre movement. In the following section, the most important empirical equations existing within the literature will be presented along with a critical evaluation of their respective advantages and disadvantages. In a pioneering paper, representative of the first effect, Bernstein (1913), having observed that the main resistance in a tyre’s movement is due to the effort of creating a rut to cross over, proposed Eq. 2.3, which was later extended by Goriatchkin (1936) to its more generalized version, that of Eq. 2.4.

0.5

p k z (2.3)

n

p k z (2.4)

The aforementioned equations were developed based on the assumptions that the soil is homogeneous and that k,n are constants in a given soil within a varied plate geometry, size, and dimension range. However, the latter assumption was found not to be valid since parameter k is highly affected by the dimensions of the plate and the soil conditions. With regards to the dimensions of the plate, it is evident that the pressure calculated using Eq. 2.3 and Eq. 2.4 is independent of the width and/or the length of the plate. Furthermore, regarding the soil conditions, the parameter k is also independent of important soil parameters such as soil moisture. Thus, use of the aforementioned equations will necessitate repetitive tests and measurements for the extraction and calculation of parameter k.

Evans (1964) experimentally studied tracked vehicles operating on clay soils and, based on his results, he proposed Eq. 2.5. Evans was the first who considered that the modulus of soil deformation k, proposed in Eq. 2.3, 2.4, consists of two different components, the first being responsible for the cohesion of the soil and the second being related to a deformation constant. In addition, Evans was the first who took into consideration the width of the wheel which had until then been omitted by every previous researcher.



az/2b



max

qq  1 e  (2.5)

M. G. Bekker, a pioneer and leading specialist in the field of Terramechanics, studied Eq. 2.4 and analyzed results with experimental data. Bekker (1957) introduced two

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different moduli for the soil; kc for the cohesive part of the soil and kφ for the

respective frictional behaviour. He developed and proposed Eq. 2.6 where the width of the rectangular plate was also included. Two basic assumptions used in the development of the current model are that: i) the soil is considered to be homogeneous and ii) a linear relationship exists between k and 1/B as k=kc/b+kφ.

n c φ k p k z b   _  _   (2.6)

Subsequently, Bekker introduced Eq. 2.7 which describes the compaction resistance for a rectangular uniformly loaded plate area.









n 1 / n c 1/ n c φ 1 W R l n 1 k b k     _{ }      (2.7)

By assessing Eq. 2.6, it is noted that in the case of cohesive clay, an increase in width b would yield a reduction in pressure, while in the case of frictional sand, a variation in width would have no effect on the respective results. Furthermore, Bekker’s equation does not take into consideration significant soil parameters like soil unit weight and moisture. Onafenko & Reece (1967) claimed that Eq. 2.6 did not take into consideration the slip and skid conditions, something which would yield significant errors and drawbacks in tyre modelling aspects. Another fundamental limitation of the aforementioned equation lies in the assumption of a constant pressure distribution across the width of the wheel, an assumption leading to significant errors for small rigid wheels (Meirrion-Griffith & Spenko, 2010). Furthermore, parameters kc and kφ

are non-invariant parameters which are highly dependent on the dimensions of the plate which has been used for the characterization of the terrain. Thus, the argument made by Bekker for global kc and kφ factors for a given soil condition is not valid. An

evaluation of Eq. 2.7 revealed that, in order to reduce the resistance due to compaction, it was more effective to increase the length of the rectangular plate rather than the width, as the latter appears in Eq. 2.7 in a higher power than the former. Finally, similarly to equations proposed by Bernstein and Goriatchkin, Eq. 2.6 adopts parameters independent of important physical soil parameters such as soil moisture, leading us to the conclusion that it can be used only for homogeneous soils.

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Later on, Bekker (1960) developed models which yielded a more accurate and representative result with regards to rigid wheels rolling response. Figure 2.2 displays a schematic representation of a rigid wheel rolling on soft terrain where the pressure acting on the circumference of the wheel is set to act radially. Bekker then proposed Eq. 2.8 in order to describe the resistance due to compaction for a towed rigid wheel, by taking into account the curvature of the wheel while at the same time omitting the contribution of tangential stresses to the lift and drag forces. Equation 2.8 was developed considering the equilibrium of forces acting on the wheel in the vertical and horizontal direction. In addition, only the front region of the contact patch of the tyre-soil interaction was set to contribute to the overall rolling resistance (from point A to point B), which – as the following sections will show – lead to significant errors and limitations in tyre traction predictions. Furthermore, Bekker proposed Eq. 2.9 for the characterization of the maximum allowed sinkage of a towed wheel.





   

_

_





  2n 2 / 2n 1   c 2n 2 / 2n 1 1/ 2n 1 c φ 1 3W R d 3 n n 1 k bk         _ _      (2.8)









  2/ 2n 1 c 3W z 3 n k bk d              (2.9)

Figure 2.2.Dynamic behaviour of a rolling rigid wheel.

However, the rolling resistance due to compaction Rc is only one of the components of the total rolling resistance of the wheel. In particular, for a towed wheel, resistance due to bulldozing conditions must also be accounted for since the terrain is

θ ω R θ0 W Z0 A B Rc σ Z

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accumulated in front of the tyre thus adding to the existing resistance. Bekker (1960) proposed Eq. 2.10 in order to estimate the bulldozing resistance Rb of a wheel rolling on a soil with a soft upper layer. Furthermore, an additional Rt resistance must be accounted for, which represents the resistance caused by the tyre deformation. Although Rt is omitted in most of the cases by the majority of the researchers, since the level of soil deformation is incomparably larger than that of the tyre deformation, Bekker & Semonin (1975) proposed Eq. 2.11for the calculation of Rt. Thus, the total rolling resistance is given in Eq. 2.12.





3





2

_

_

2 2 b c γ bsin a φ πt γ 90 φ cπt _φ R 2zcK γz K ct tan 45 2 2sinαcosφ 540 180      _  _    (2.10)









2 t gr t R _3.581bD p ε 0.0349α sin2α _/ α D 2δ (2.11) tot c b t R R R R (2.12)

In addition, Bekker (1960) considered the response of a pneumatic tyre interacting with a soft soil in which case a flat contact patch approximation was used and tyre deflection was set to affect the rolling resistance due to compaction; following that, Eq. 2.13 was proposed.





1 n 1 n c n c gr k R b p n 1 k b       _ _ _ _ _  _  _  _  _ _   _{ } _ (2.13)

Bekker introduced numerous vehicle performance metrics, such as Thrust (H) and Drawbar Pull (DP), based on which the locomotion of a wheel can be assessed. Reece observed that kc and kφ of Bekker’s model have variable dimensions and their value is dependent on the exponent n; thus, the improved Reece “ model I ” was proposed (Reece, 1965; Onafenko & Reece, 1967). This model, as illustrated in Eq. 2.14, utilizes two parameters with constant dimensions of Pa and Pa/m respectively.



1 2



n z p k k b b     _{ }    (2.14)

Following that, Reece (1965) based on, Terzaghi’s (1944) and Meyerhofs’s (1951) bearing capacity theories for plasticity, used soil mechanics to examine the soil failure underneath a strip. Reece proposed Eq. 2.15, which has the advantage of a sound

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- 16 -

theoretical basis and two dimensionless moduli, kc’ and kφ’ (Reece model II).

Furthermore, Reece noted that kc’ and kφ’are invariant parameters for given soil

conditions and do not depend on the dimensions of the plate. However, as it was highlighted by Upadhyaya et al. (1993), Reece’s proposed equation provided improved predictions in terms of pressure-sinkage compared to Eq. 2.6, but kc’ and kφ’remained parameters dependent on the dimensions of the plate. Furthermore, Wills

(1963) and Lyasko (2010a) proved that there is no linear relation between k and b, since the influence of plate dimensions on kc’ and kφ’ was significant.





 

n ' ' c φ z p ck γbk b   (2.15)

It is evident that for frictional sand, with significantly no cohesion, the first term in Eq. 2.15 will be omitted while for cohesive clay the second term will exhibit negligible levels. Reece’s relationship is regarded to be a significant improvement in the field of Terramechanics and a major contribution to the overall tyre-soil interaction. Extensive experimental work conducted by Wills (1963) confirmed the value of Reece’s Eq. 2.15 despite the fact that his model varied from Eq. 2.4 only in its response regarding the width b. By assessing Eq. 2.15 and by increasing plate width b loaded on cohesive clay, a linear increase in pressure is caused. On the contrary, in the case of frictional sand a variation in width b would yield a pressure proportional to b/bn. Finally, it is apparent that significant soil parameters, such as soil moisture and hardpan depth (the thickness of the upper layer of the soil which can be deformed under loading) are not taken into consideration.

Yousesef & Ali (1982) proposed Eq. 2.16 where K1 and K2 are soil shear values and a

and β are dimensionless geometrical constants. Equation 2.16 was validated with many penetration tests and following that, a direct comparison with Bekker’s coefficients was found. It should be mentioned once more that the parameters used in Eq. 2.16 are non-invariant parameters, since direct correlation with Bekker’s parameters exists, a fact which necessitates continuous measurements and in situ tests regarding the calculation of their values. It is also worth repeating that the above-stated pressure sinkage equations can represent mostly homogeneous terrains (no hardpan depth). For non-homogeneous terrains, different approaches are available to account for the inherent behavior of the different layers of the soils. An example of

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such an expression is proposed by Wong (2001) with Eq. 2. 17; suitable for representing organic terrains (muskeg).





n n 1 2 z p K a B K B       _{  }    (2.16) 2 m p h 4m z p k z D   (2.17)

All the equations mentioned in the current section contain non-invariant parameters, highly dependable on the physical characteristics of soils and applicable only to homogeneous terrains where a monotonic sinkage decrease with a ground pressure increase can be considered. Therefore, the values for these parameters (involved in all the above mathematical models) cannot be used beyond the soil conditions for which they have been measured without additional experimental testing.

2.2.3 Shear Stress-Shear Displacement

The second target relationship deals with the relationship between shear stress andshear displacement. Shear stress is applied on the terrain surface via a vehicle’s running gear, causing in this way the development of thrust and its associated slip characteristics. The limits of soil strength prior to terrain failure are crucial for the development of such models. The following section reviews the most prevalent equations in the field of Terramechanics, used for capturing accurately the off road tyre-soil interaction. In this section the soil shear stress response is divided into three main categories. The word “hump” (peak) will be utilised with the same meaning as by Wong (2010).

The first category tends to capture the behavior of soils similar to sands, saturated clay and fresh snow. These soils do not exhibit a “hump” of maximum shear stress, Fig.2.3(a), thus by increasing the shear displacement j, the shear stress reaches a maximum value and remains constant with further increase in j . Janosi & Hanamoto (1961) proposed Eq. 2.18 in their effort to capture the response of the above stated category and remains until now one of the most widely used and adopted equations. In Eq. 2.18 the term in the first brackets represents the maximum shear stress; this mathematical relationship expresses the maximum shear strength of a soil specimen and was initially proposed by a French physicist, Charles Augustin de Coulomb, and further developed by a civil engineer, Christian Otto Mohr, leading to the

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well-- 18 well--

established final Mohr-Coulomb failure criterion. This postulates that the material will fail at a point where the shear stress reaches a maximum value. Furthermore, it should be noted that an incorrect value for the shear deformation modulus (Kx) may

lead to incorrect shear stress values which in return will create an unrealistic and unreliable tyre-soil interaction model. Available experimental data in the literature (e.g. Wong, 2001) suggest that Kx is highly dependent on ground pressure; however,

an accurate empirical and/or analytical relationship is yet to be determined. Nevertheless, Lyasko (2010c) having conducted numerous tests for tracked vehicles in various soil conditions stated that Kxis a function of the internal friction angle and

for 150 ≤ φ0 ≤ 400 he proposed Eq. 2.19.





x x j K τ( )  c p(θ) tan φ 1 e _   _   (2.18) 0 0 K 0.0039 0.055 (2.19)

Wong (1979) and Wong & Preston-Thomas (1983) proposed Eq. 2.20 for soils which exhibit a “hump” of maximum shear stress and then by further increasing the shear displacement the shear stress continuously decrease Fig.2.3 (b). Equation 2.20 was validated and close agreement with experimental results was observed.

w

(1 j/K ) w

(c p tan )( j / K )e 

    (2.20)

With regards to the third category of shear stress response, the shear stress exhibits a “hump” which then with further increase in the shear displacement, decreases to a constant value. This trend is illustrated in Fig.2.3(c). Oida (1979) based on work of Pokrovski’s and of Kacigin and Guskov (1968), proposed a mathematical expression representative of this category, which was further modified and established in its final form by Wong (1983) as in Eq. 2.21. However, as Lyasko (2010d) states, Kr and Kw

used in Eq. 2.20 and 2.21 are non-invariant soil plate parameters and can be determined only experimentally.











j/ Kw





1 j/ Kw



r r

c p tan φ K 1 1/ K 1 1/ e 1 e 1 e 

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Figure 2.3. Soils’ Shear Stress response for three different types of soil, Wong(2001).

It should be highlighted that changes in soil conditions (e.g. moisture content variation) influence tyre and consequently vehicle performance to a greater extent, compared to changes in tyre inflation/loading conditions and/or its size and dimensions (Upadhyaya et al., 1989). In addition, it can be argued that it is impossible to carry out identical tests under the same conditions (e.g. identical moisture content) due to a variety of soil parameters such as the fluctuation of the weather. Following that, an increase in the number of runs would be a prerequisite in terms of reducing the overall experimental error. Therefore, it should be noted that soil characteristics (e.g. shear strength) must be defined in an analytical manner, and not via empirical equations, with the use of invariant soil parameters such as, cohesion and internal friction angle.

Occasionally, use of the above empirical equations may lead to incorrect results and ambiguous answers regarding a vehicle’s tranversability. For instance, having studied a rubber belted tractor on three different soils for three different belt widths, Zoz (1997) suggests that with the use of wider belts there is an increase in the traction performance of the tractor. This is contrary to the findings of a similar study, conducted by Bashford & Kocher (1999) which argues that the tyres with narrower belts provide the optimum traction performance. Yet, Upadhyaya et al. (2001), suggest that the belt width does not significantly affect the tractive performance of a vehicle. It is evident, that in the above-mentioned studies, use of an empirical relationship led to three different conclusions associated with the width of the belt, regarding the overall tractive performance of a vehicle on deformable soils; a fact which highlights the assertion that empirical equations may lead to erroneous or ambiguous findings.

Analytical and finite element modelling of the dynamic interaction between off-road tyres and deformable terrains